The invention relates to a method and a recording medium for measuring a thickness profile and a refractive index, more particularly, to a method and a recording medium for measuring three-dimensional thickness profile and refractive index of transparent dielectric thin-film with some patterns or not, which is fabricated in the semiconductor and related industrial field, using white-light scanning interferometry.
It is well known that white-light scanning interferometry is useful for measuring three-dimensional profile of patterns such as semiconductor patterns with the size of a few microns or submicrons.
White-light scanning interferometry also has advantages of being capable of measuring a rough surface or a measurement surface having high step height with high resolution by overcoming 2xcfx80 phase ambiguity of the conventional phase-shifting interferometry.
The fundamental principle used for measurement using white-light scanning interferometry is utilizing short coherence length characteristics of white-light.
And the principle of the measurement method is that an interference signal appears only when reference light and measurement light, which are split by a beam splitter, have nearly the same optical path difference.
Therefore, observing the interference signal at a measurement point within measurement region, while moving a measurement object in a direction of light axis (optical axis) at minute intervals, for example, a few nanometers, a short interference signal appears at points where each point has the same optical path difference as a reference mirror.
The three-dimensional profile can be measured by extracting points of these interference signals at all measurement points within measurement region.
These researches for white-light scanning interferometry have been actively developed since 1980s.
Generally, white-light interference signal can be expressed as following equation.
I=IDC+xcex3 cos xcfx86xe2x80x83xe2x80x83[Equation 1]
In the above equation 1, I is white-light interference signal which is represented by average light intensity of interference signal IDC, visibility function xcex3, and cosine function.
As amplitude function of interference signal which varies slowly compared to the interference signal""s phase variation represented as cosine function, visibility function xcex3 is generally expressed as symmetric function such as Gaussian function or polynomial function format and also has many synonyms such, for example, as envelope function, modulation, signal variance, and so on.
In the ideal interference system, the maximum interference signal is occurred at a point, where the optical path difference between reference plane and measurement point is zero. And the point becomes height of the measurement point.
However, there are lots of errors caused by a number of error sources in the actual interference system. Those error sources are phase variation by reflection at a measurement point, an influence of actuation interval in a direction of light axis, frequency characteristics of light source, and so on
For this reason, the primary research field for interference measurement method is improving correctness and repetitiousness of measurement value by complementing above mentioned error sources.
Generally, the development of these researches can be classified into three methods.
The first method is determining a point, where visibility of an interference signal is maximum, as measurement value.
The second method is determining measurement value by slope of phase graph per frequency of an interference signal.
The third method is utilizing phase-shifting method and then determining a measurement point.
The detailed and numerous descriptions are disclosed in the article by Kang Min Ku: The study on the surface profile measurement algorithm using white-light scanning interferometry, KAIST, Ph D. thesis (1999).
And one of the commercially widely used methods is phase analysis per frequency (U.S. Pat. No. 5,398,113, Mar. 14, 1995) proposed by Peter De Groot.
The method greatly improved the disadvantages of the conventional methods using visibility function of an interference signal.
Above mentioned disadvantages are including lots of time for calculation, noise sensitive calculation algorithm, influence by frequency of light source, need for moving distance of two or more signal values per one interference signal period to precisely extract visibility function, and so on.
The white-light source was represented as sum of several wavelengths within uniform bandwidth, and the following relationship [equation 2] between profile information and phase variation per frequency was given by considering an interference signal acquired from white-light source as incoherent superposition.
"PHgr"((k)=2kh+xcfx86Oxe2x80x83xe2x80x83[Equation 2]
In the above equation 2, k means propagation vector or angular wave number and mathematically represented as 2xcfx80/xcex.
And here, xcex means optical wavelength of light source and h means profile value of a measurement point and is represented as relative distance value to reference mirror. Also xcfx86o is phase variation by surface reflection and is represented by Fresnel equation. And in the case of measuring surface composed of only a single material, the value of xcfx86o is constant through the entire measurement region.
It is understood that phase variation value is represented as a simple equation for the propagation vector k and the slope is profile value of the object to be measured from the above equation 2.
However, these all measurement algorithms have the disadvantages of being applied when reflection is occurred only in the surface and the measurement surface is composed of opaque material.
It is generally known that an interference signal is distorted by multiple reflection phenomena in the thin film when dielectric material of a few nanometers or micrometers is deposited on the surface of the measurement plane.
However, in the case of applying a method using conventional visibility function to the interference signal or a method adopting phase-shifting algorithm, the interference signal appears unsymmetrically by the effect of thin film and the maximum visibility may be appeared at a position where the optical path difference is not zero.
And the position can cause measurement errors.
Also, the method using phase variation per frequency leads to the variation of value of above equation 2 according to thickness of the measurement point.
Therefore, above equation 2 is not represented as a simple equation according to the propagation vector any more, and represented as a complex function format, where the nonlinear elements are added to the linear elements expressed as an equation of the first degree.
Thus the conventional algorithms have many problems causing serious measurement errors.
A method and a recording medium for measuring three-dimensional thickness profile and refractive index of transparent dielectric-thin-film with some patterns or not, which is fabricated in the semiconductor and related industrial field, using white-light scanning interferometry is provided.
A method for measuring a thickness profile using white-light scanning interferometry in optical system includes the following steps.
(a) extracting a phase graph by acquiring an interference signal and performing Fourier transform;
(b) extracting a mathematical phase graph through modeling of a measurement object; and
(c) measuring a profile value and a thickness value by applying an optimization technique to an error function determined by using phase values which is acquired from said step (a) and step (b).
Preferably, wherein said step (c) further includes the following steps.
(d) setting up an error function by using said phase values which is acquired from said step (a) and said step (b);
(e) determining search start point by setting up search region at an arbitrary measurement point;
(f) calculating an error function value and a convergence point where said error function value becomes minimum, for each arbitrary search start point;
(g) setting up said convergence point, which has minimum error function value out of a plurality of search start points for said arbitrary measurement point, as measurement value; and
(h) determining whether said measurement values are set for all measurement points, and repeatedly performing said step (e), said step (f), and said step (g) if said measurement values are not set.
Preferably, wherein said error function of said step (d) has characteristics of using said phase value "PHgr"m(k) acquired from said step (a) and said phase value "PHgr"c(k) of a model acquired from said step (b) and expressed as following equation.       χ    2    =            ∑      k        ⁢          xe2x80x83        ⁢                  [                                            Φ              m                        ⁢                          (              k              )                                -                                    Φ              c                        ⁢                          (                                                k                  ;                  h                                ,                d                            )                                      ]            2      
Where, k and h means a propagation vector and a configuration value, respectively and d is thickness value of thin film constituting thin film structure.
More preferably, wherein said optimization technique, which is utilized to detect said profile and said thickness values, characterizes utilizing nonlinear least square method in said step (c).
Preferably, wherein said optimization technique, which is utilized to detect said profile and said thickness values, characterizes utilizing nonlinear least square method of Levenberg-Marquardt in said step (c).
A recording medium readable with computer in which program for measuring a thickness profile using white-light scanning interferometry is provided. The program includes the following steps.
(a) extracting a phase graph by acquiring an interference signal and performing Fourier transform;
(b) extracting a mathematical phase graph through modeling of a measurement object; and
(c) measuring a profile value and a thickness value by applying an optimization technique to an error function determined by using phase values which is acquired from said step (a) and said step (b).
A method for measuring a refractive index using white-light scanning interferometry in optical system is provided. The method includes the following steps.
(a) extracting a phase graph by acquiring an interference signal and performing Fourier transform;
(b) extracting a mathematical phase graph through modeling of a measurement object; and
(c) measuring a refractive index by applying an optimization technique to an error function determined by using phase values which is acquired from said step (a) and said step (b).
Preferably, wherein said step (c) further includes the following steps.
(d) setting up an error function by using said phase value which is acquired from said step (a) and said step (b);
(e) determining search start point by setting up search region at an arbitrary measurement point;
(f) calculating an error function value and a convergence point where said error function value becomes minimum, for each arbitrary search start point;
(g) setting up said convergence point, which has minimum error function value out of a plurality of search start points for said arbitrary measurement point, as a measurement value; and
(h) determining whether said measurement values are set up for all measurement points, and repeatedly performing said step (e), said step (f), and said step (g).
More preferably, wherein said error function of said step (d) has characteristics of using said phase value "PHgr"m(k) acquired from said step (a) and said phase value "PHgr"c(k) of a model acquired from said step (b) and expressed as following equation.       χ    2    =            ∑      k        ⁢          xe2x80x83        ⁢                  [                                            Φ              m                        ⁢                          (              k              )                                -                                    Φ              c                        ⁢                          (                              k                ;                                  N                  i                                            )                                      ]            2      
Where, k means a wave number and Ni means a refractive index of thin film constituting thin film structure.
Preferably, wherein said optimization technique, which is utilized to detect said refractive index, characterizes utilizing nonlinear least square method in said step (c).
More preferably, wherein said optimization technique, which is utilized to detect said refractive index, characterizes utilizing nonlinear least square method of Levenberg-Marquardt in said step (c).
A recording medium readable with computer in which program for measuring a refractive index using white-light scanning interferometry is provided. The program includes the following steps.
(a) extracting a phase graph by acquiring an interference signal and performing Fourier transform;
(b) extracting a mathematical phase graph through modeling of a measurement object; and
(c) measuring a refractive index by applying optimization technique to said error function determined by using said phase value which is acquired from said step (a) and said step (b).